If you have ssh access to the server you can do this:
cd /var/discourse
./launcher enter app
u=User.find_by_email('old@email.address')
u.email='new@email.address'
u.save;
exit
exit
But I just confirmed that for a regular user, no email is sent to the current email address. Perhaps your account is both an admin and a moderator? (I didn’t test if moderator accounts can be changed without notifying the old address).
A notification email is sent to the old address, but the confirmation email is sent only to the new address.